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What is the coordination number of each sphere in (a) a simple cubic cell, (b) a body-centered cubic cell, and (c) a face-centered cubic cell? Assume the spheres are all the same.

Short Answer

Expert verified
(a) 6, (b) 8, (c) 12

Step by step solution

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01

Understanding Coordination Number

The coordination number is defined as the number of nearest neighbors that surround a given atom in a crystal lattice.
02

Simple Cubic Cell Coordination Number

In a simple cubic (SC) cell, each sphere is located at the corners of a cube. Each sphere touches one sphere on each of its 6 faces, thus it has a coordination number of 6.
03

Body-Centered Cubic Cell Coordination Number

In a body-centered cubic (BCC) cell, there is one sphere at the center of the cube and one at each corner. The sphere at the center touches all 8 corner spheres, and each corner sphere touches the central sphere plus its nearest neighbors on the corners. Therefore, the coordination number is 8.
04

Face-Centered Cubic Cell Coordination Number

In a face-centered cubic (FCC) structure, atoms are located at the corners and the centers of each face of the cube. Each corner atom is surrounded by 12 other atoms: 4 on the same face, 4 on adjacent faces above, and 4 below. Thus, the coordination number is 12.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simple Cubic Cell
A simple cubic (SC) cell is one of the basic crystal lattice structures you can find in solid materials. In this structure, each atom is situated at the corners of the cube. This might seem quite straightforward, but there's more to it than meets the eye! Each atom in a simple cubic cell has a coordination number of 6. This means each atom is in direct contact with six other atoms around it. To visualize this, imagine that each corner atom is connected to one atom on each of the six cube faces. The coordination number indicates how densely packed the atoms are and influences the properties of the material. Simple cubic cells are not very common in elements because the packing efficiency is quite low, essentially leaving lots of unused space within the structure. Understanding this can help explain why materials with simple cubic cells might have lower densities than others with different atomic arrangements.
Body-Centered Cubic Cell
The body-centered cubic (BCC) cell is a bit more intricate. Picture a cube, but in addition to atoms at each corner, there is one atom right in the center of the cube. This central atom links the corner atoms together. Because of this arrangement, the coordination number of a BCC structure is 8. Let's break that down:
In the BCC structure, the central atom directly touches the 8 corner atoms. Also, each corner atom interacts with the center atom and has some associations with neighboring corner atoms as well.

This somewhat tighter packing compared to the simple cubic cell results in better space utilization. Imagine cramming more atoms into the same cube – this is not only a more efficient use of space but also impacts the material's mechanical properties, often making it stronger than materials with a simple cubic structure. Metals such as iron and chromium commonly exhibit a BCC arrangement.
Face-Centered Cubic Cell
Face-centered cubic (FCC) cells take atomic dense packing to the next level. Here, atoms are not only at each corner of the cube but also at the center of every face of the cube. Because of this arrangement, each atom is closely packed with its neighbors, resulting in a coordination number of 12. To better envision this, think of each corner atom being surrounded by 4 atoms on the same face, 4 on an adjacent face above, and 4 below.

The FCC structure, with its high coordination number, is all about maximizing space efficiency. This accounts for why many metals, such as aluminum, copper, and gold, exhibit such a structure – it grants them excellent ductility and malleability. These properties make FCC-structured materials favorable in various applications, like electronics and jewelry. It's fascinating to see how slight changes in atomic arrangement can lead to vastly different physical properties!

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